Computation of Hardy-Sobolev inequality

1. The case p=2+2/N

1.1 Define U

We shall denote U1 as the function. Denote r=|x’| and t=x_N. Here n is the dimension.

Clear[t,r,U1,p,c,n];
p=2+2/n;s=n-(n-2)p/2;
U1=(((2n))^(1/(p-2)))(t/((((((1+t))^2)+(r^2)))^(n/2)));

Verify  that U satisfies the equation

LapU=Simplify[Laplacian[U1,{r,t}]+(((n-2))/r)D[U1,r]];
tsU1p=FullSimplify[PowerExpand[(t^(-s))((( U1))^(p-1))]];
Simplify[-LapU/tsU1p]

1.2 Compute I1 and I2

We shall use (Subscript[ω, n-2]) denote the volume of the sphere (S^(N-2)).

nablaU1=Simplify[D[U1,r]^2+D[U1,t]^2];
t1nablaU1=FullSimplify[(Subscript[ω, n-2])Integrate[t nablaU1 r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=2]];
t1dx1U1=FullSimplify[(Subscript[ω, n-2])Integrate[t (D[U1,r])^2 r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=2]/(n-1)];
ts1U1p=FullSimplify[(Subscript[ω, n-2])Integrate[t^(-s+1) (U1)^(p) r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=2]];
I1=Simplify[-t1nablaU1+2t1dx1U1];
I2=ts1U1p;FullSimplify[I1-(-2(n-1)+s)/(n-1)/(p)I2]

1.3 Compute I3 and I4

a11=(Subscript[ω, n-2])Integrate[t^2(nablaU1)r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3];
a13=(Subscript[ω, n-2])Integrate[t^2(D[U1,r])^2r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3]/(n-1);
a14=(Subscript[ω, n-2])Integrate[r^2(nablaU1)r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3]/(n-1);
I3=FullSimplify[-1/2a11+1/6a14+3a13];
b11=(Subscript[ω, n-2])Integrate[t^(2-s)(U1)^(p)r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3];
b12=(Subscript[ω, n-2])Integrate[t^(-s)(U1)^(p)r^(n),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3]/(n-1);
I4=FullSimplify[1/2b11-1/6b12];
FullSimplify[I3+2/(p)I4]

1.4 Compute I5 and I6

Z1=D[U1,r]/r;
nablaZ1=D[Z1,r]^2+D[Z1,t]^2;
I5coeffA2=2/(n^2-1)(Subscript[ω, n-2])Integrate[t^2r^4 nablaZ1 r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3];
I5coeffA1=8/(n^2-1)(Subscript[ω, n-2])Integrate[t^2(r D[U1,r]-r^2D[U1,r,r])Z1 r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3];
I6coeffA2=-p(p-1)/(n^2-1)(Subscript[ω, n-2])Integrate[t^(2-s)r^4U1^(p-2)Z1^2r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3];
DN=(n^2-1)/8FullSimplify[I5coeffA1^2-4(I3+2/(p)I4)(I5coeffA2+2/p I6coeffA2)]
N[Solve[DN/((Subscript[ω, n-2]))^2==0,Assumptions->n>=3]]

2. The case p=2+4/N

2.1 Define U

Clear[t,r,U2,p,n];
p=2+4/n;s=n-(n-2)p/2;
U2=((((2+n)n))^(1/(p-2))) (t/(((1+(t^2)+(r^2)))^(n/2)));

Verify  that U satisfies the equation

LapU=Simplify[Laplacian[U2,{r,t}]+(((n-2))/r) D[U2,r]];
tsU2p=FullSimplify[PowerExpand[(t^(-s))((( U2))^(p-1))]];
Simplify[PowerExpand[-LapU/tsU2p]]

2.2 Compute I1 and I2

nablaU2=Simplify[D[U2,r]^2+D[U2,t]^2];
t1nablaU2=FullSimplify[(Subscript[ω, n-2])Integrate[t nablaU2 r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=2]];
t1dx1U2=FullSimplify[(Subscript[ω, n-2])Integrate[t (D[U2,r])^2 r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=2]/(n-1)];
ts1U2p=FullSimplify[(Subscript[ω, n-2])Integrate[t^(-s+1) (U2)^(p) r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=2]];
I1=Simplify[-t1nablaU2+2t1dx1U2];
I2=ts1U2p;FullSimplify[I1-(-2(n-1)+s)/(n-1)/(p)I2]

2.3 Compute I3 and I4

a21=(Subscript[ω, n-2])Integrate[t^2(nablaU2)r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3];
a23=(Subscript[ω, n-2])Integrate[t^2(D[U2,r])^2r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3]/(n-1);
a24=(Subscript[ω, n-2])Integrate[r^2(nablaU2)r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3]/(n-1);
I3=FullSimplify[-1/2a21+1/6a24+3a23];
b21=(Subscript[ω, n-2])Integrate[t^(2-s)(U2)^(p)r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3];
b22=(Subscript[ω, n-2])Integrate[t^(-s)(U2)^(p)r^(n),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3]/(n-1);
I4=FullSimplify[1/2b21-1/6b22];
FullSimplify[I3+2/(p)I4]

2.4 Compute I5 and I6

Z2=D[U2,r]/r;
nablaZ2=D[Z2,r]^2+D[Z2,t]^2;
I5coeffA2=2/(n^2-1)(Subscript[ω, n-2])Integrate[t^2r^4 nablaZ2 r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3];
I5coeffA1=8/(n^2-1)(Subscript[ω, n-2])Integrate[t^2(r D[U2,r]-r^2D[U2,r,r])Z2 r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3];
I6coeffA2=-p(p-1)/(n^2-1)(Subscript[ω, n-2])Integrate[t^(2-s)r^4U2^(p-2)Z2^2r^(n-2),{t,0,Infinity},{r,0,Infinity},Assumptions->n>=3];
DN=(n^2-1)/8FullSimplify[I5coeffA1^2-4(I3+2/(p)I4)(I5coeffA2+2/p I6coeffA2)]